Algebraic properties of lattices of quotients of nite posets are considered. Using the known duality between the category of all nite posets together with all order-preserving maps and the category of all nite distributive (0; 1)-lattices together with all (0; 1)-lattice ho-momorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of...

In this paper we prove that if G is a group for which there are k non-Frattini chief factors isomorphic to characteristically simple A, then has normal section C/R the direct product of minimal subgroups G/R A. This significant extension notion crown factors.

In this note we show that if two critical algebras generate the same congruence-permutable variety, then the varieties generated by their proper sections also coincide.

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...